Solving boundary value problems using critical point theory

Abstract

In this study, we use minimax methods introduced by Rabinowitz [33] to show the existence of nontrivial solutions to certain boundary value problems. A nonlinear 2n^th order difference equation on a finite interval with two point boundary conditions is shown to have a nontrivial solution. Assumptions are also given to show the existence of multiple solutions to this problem. This problem is then extended from a finite interval to the integers, where boundary conditions are given at infinity. Existence of solutions to this problem are shown, with additional conditions given to guarantee an unbounded sequence of homoclinic orbits to this difference equation. Finally, a second order, nonlinear delta-nabla equation with a parameter is considered on a finite time scale interval and shown to have multiple solutions.